Question: Factor the following expression: $7$ $x^2+$ $41$ $x+$ $30$
Explanation: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(7)}{(30)} &=& 210 \\ {a} + {b} &=& & & {41} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $210$ and add them together. The factors that add up to ${41}$ will be your ${a}$ and ${b}$ When ${a}$ is ${6}$ and ${b}$ is ${35}$ $ \begin{eqnarray} {ab} &=& ({6})({35}) &=& 210 \\ {a} + {b} &=& {6} + {35} &=& 41 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {7}x^2 +{6}x +{35}x +{30} $ Group the terms so that there is a common factor in each group: $ ({7}x^2 +{6}x) + ({35}x +{30}) $ Factor out the common factors: $ x(7x + 6) + 5(7x + 6) $ Notice how $(7x + 6)$ has become a common factor. Factor this out to find the answer. $(7x + 6)(x + 5)$